Classical algorithms, correlation decay, and complex zeros of partition functions of quantum many-body systems

In this paper, we present a quasi-polynomial time classical algorithm that estimates the partition function of quantum many-body systems at temperatures above the thermal phase transition point. It is known that in the worst case, the same problem is NP-hard below this point. Together with our work, this shows that the transition in the phase of a quantum system is also accompanied by a transition in the hardness of approximation. We also show that in a system of n particles above the phase transition point, the correlation between two observables whose distance is at least log(n) decays exponentially. We can improve the factor of log(n) to a constant when the Hamiltonian has commuting terms or is on a 1D chain. The key to our results is a characterization of the phase transition and the critical behavior of the system in terms of the complex zeros of the partition function. Our work extends a seminal work of Dobrushin and Shlosman on the equivalence between the decay of correlations and the analyticity of the free energy in classical spin models. On the algorithmic side, our result extends the scope of a recent approach due to Barvinok for solving classical counting problems to quantum many-body systems.Comment: 54 pages, 4 figure

There is entanglement in the primes

Large series of prime numbers can be superposed on a single quantum register and then analyzed in full parallelism. The construction of this Prime state is efficient, as it hinges on the use of a quantum version of any efficient primality test. We show that the Prime state turns out to be very entangled as shown by the scaling properties of purity, Renyi entropy and von Neumann entropy. An analytical approximation to these measures of entanglement can be obtained from the detailed analysis of the entanglement spectrum of the Prime state, which in turn produces new insights in the Hardy-Littlewood conjecture for the pairwise distribution of primes. The extension of these ideas to a Twin Prime state shows that this new state is even more entangled than the Prime state, obeying majorization relations. We further discuss the construction of quantum states that encompass relevant series of numbers and opens the possibility of applying quantum computation to Arithmetics in novel ways.Comment: 30 pages, 11 Figs. Addition of two references and correction of typo

Fisher zeros and persistent temporal oscillations in nonunitary quantum circuits

Funding: UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/R031924/1.We present a quantum circuit with measurements and postselection that exhibits a panoply of space- and/or time-ordered phases from ferromagnetic order to spin-density waves to time crystals. Unlike the time crystals that have been found in unitary models, those that occur here are incommensurate with the drive frequency. The period of the incommensurate time-crystal phase may be tuned by adjusting the circuit parameters. We demonstrate that the phases of our quantum circuit, including the inherently nonequilibrium dynamical ones, correspond to complex-temperature equilibrium phases of the exactly solvable square-lattice anisotropic Ising model.Publisher PDFPeer reviewe

Phase Transitions, Critical Phenomena, and Correlation Functions in the 2D Ising Model and its Applications to Quantum Dynamics: A Tensor Network Approach

This thesis explores several aspects of the 2D Ising Model at both real and complex temperatures utilizing tensor network algorithms. We briefly discuss the importance of tensor networks in the context of forming efficient representations of wavefunctions and partition functions for quantum and classical many-body systems respectively, followed by a brief review of the tensor network renormalization algorithms to compute the one point and two point correlation functions. We use the Tensor Renormalization Group (TRG) to study critical phenomena and examine feasibility of accurate estimations of universal critical data for three critical points for three critical points in two dimensions -- the critical points for the isotropic and the anisotropic square lattice Ising models, and the Yang-Lee critical endpoint. The latter two exhibit appreciable corrections to scaling, making a clear case for uniform convergence in bond dimension apparent in our results. We are able to reproduce exactly known values to within 1 percent with modest effort of bond dimension 28. We analytically continue into the complex temperature plane to identify and study novel phases and phase transitions of the 2D isotropic as well as the 2D anisotropic Ising model. Regions of the phase space that lie in the complex coupling plane cannot be studied using standard Quantum Monte Carlo techniques (phase problem for complex coupling). Tensor networks provides us with a powerful tool to analyze and study regions of phase space in complex coupling planes. Evidence from tensor network renormalization techniques for the infinte 2D isotropic Ising model suggest the presence of only the standard paramagnetic (PM) and the ferromagnetic (FM) phases in the complex temperature plane. On the other hand, evidence from magnetization using tensor network renormalization techniques for the 2D anisotropic Ising model, and also numerical evidence based on the exact Onsager solution for the infinte 2D anisotropic Ising model suggests the presence of novel phases in certain regions of the complex temperature $\tanh(\beta)$ plane that exhibits quasi-long range modulated correlations. Phase transitions from the paramagnetic (PM) phase to these novel phases (which we label as non-ferromagnet or NFM phase) exhibit a one-sided square root singularity akin to a commensurate-incommensurate phase transition. In this thesis, we also present a quantum circuit with measurements and post-selection that exhibits a panoply of space- and/or time-ordered phases. These phases can range from ferromagnetic order to spin-density waves to time crystals. The corresponding time crystals for the quantum circuit presented in this thesis are incommensurate with the drive frequency, a behavior that is a deviation from the time crystal that have been found in unitary circuits. The period of the incommensurate time-crystal phase can be tuned by adjusting the parameters for the quantum circuit. We demonstrate that these novel phases, including the inherently non-equilibrium dynamical phases, correspond to the complex-temperature equilibrium phases of the exactly solvable infinite square-lattice anisotropic Ising model with doubly periodic boundary conditions. We also present a quantum circuit and briefly discuss special dual unitary points in the complex temperature plane for the square-lattice isotropic Ising Model and its implications in quantum information theory

Decay properties of spectral projectors with applications to electronic structure

Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential off-diagonal decay ("nearsightedness") for the density matrix of gapped systems at zero electronic temperature in both orthogonal and non-orthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for non-metallic systems. We further discuss the case of density matrices for metallic systems at positive electronic temperature. A few other possible applications are also discussed.Comment: 63 pages, 13 figure

Quantum many-body systems in thermal equilibrium

The thermal or equilibrium ensemble is one of the most ubiquitous states of matter. For models comprised of many locally interacting quantum particles, it describes a wide range of physical situations, relevant to condensed matter physics, high energy physics, quantum chemistry and quantum computing, among others. We give a pedagogical overview of some of the most important universal features about the physics and complexity of these states, which have the locality of the Hamiltonian at its core. We focus on mathematically rigorous statements, many of them inspired by ideas and tools from quantum information theory. These include bounds on their correlations, the form of the subsystems, various statistical properties, and the performance of classical and quantum algorithms. We also include a summary of a few of the most important technical tools, as well as some self-contained proofs.Comment: 42 Pages + References, 7 Figures. Parts of these notes were the basis for a lecture series within the "Quantum Thermodynamics Summer School 2021" during August 2021 in Les Diablerets, Switzerlan